TITLE: Why is the determinant of the 0x0 matrix equal 1?
QUESTION [16 upvotes]: I don't understand why the determinant of the 0x0 matrix equals 1.

REPLY [15 votes]: The only linear map $\{0\}\to\{0\}$ is the identity map and the identity map has determinant $1$, because this is a required property of the determinant.
From a slightly different point of view, since the basis of $\{0\}$ is empty, the representative matrix of the unique endomorphism is the empty matrix (which is invertible and its inverse is, of course, the empty matrix). Invertible matrices should have nonzero determinant, shouldn't they?